The Brauer dimension of a field F is defined to be the least number n such that index(A) divides period(A)^n for every central simple algebra A defined over any finite extension of F. One can analogously define the Brauer-p-dimension of F for p, a prime, by restricting to algebras with period, a power of p. The 'period-index' questions revolve around bounding the Brauer (p) dimensions of arbitrary fields. In this talk, we look at the period-index question over complete discretely valued fields in the so-called 'bad characteristic' case. More specifically, let K be a complete discretely valued field of characteristic 0 with residue field k of characteristic p > 0 and p-rank n (= [k:k^p]). It was shown by Parimala and Suresh that the Brauer p-dimension of K lies between n/2 and 2n. We will investigate the Brauer p-dimension of K when n is small and find better bounds. For a general n, we will also construct a family of examples to show that the optimal upper bound for the Brauer-p-dimension of such fields cannot be less than n+1. These examples embolden us to conjecture that the Brauer p-dimension of K lies between n and n+1. The proof involves working with Kato's filtrations and bounding the symbol length of the second Milnor K group modulo p in a concrete manner, which further relies on the machinery of differentials in characteristic p as developed by Cartier. This is joint work with Bastian Haase.

The Polya-Vinogradov inequality, an upper bound on character sums proved a century ago, is essentially optimal. Unfortunately, it's also not so useful in applications, since it's nontrivial only on long sums (while in practice one usually needs estimates on sums which are as short as possible). The best tool we have to handle shorter sums is the Burgess bound, discovered in 1957; this is generally considered to supersede Polya-Vinogradov, both because its proof is "deeper" (building on results from algebraic geometry) and because it is more applicable. In this talk I will introduce and motivate both of these bounds, and then describe the unexpected result (joint with Elijah Fromm, Williams '17) that even a tiny improvement of the (allegedly weaker) Polya-Vinogradov inequality would imply a major improvement of the (supposedly superior) Burgess bound. I'll also discuss a related connection between improving Polya-Vinogradov and the classical problem of bounding the least quadratic nonresidue (joint with Jonathan Bober, University of Bristol).

Students from two projects of the Rice Geometry Lab will be doing a poster presentation to showcase what they have done this semester. Prof. Roy and Prof. Li have been directing teams of undergraduate students this semester on the projects 'Music and Geometry’ and 'Understanding the works by Nash on isometric embeddings'. Graduate students, Yikai Chen and Xian Dai, have been helping the teams out. Come take a look at what some of your fellow undergraduate students are doing! We will be looking for interested undergraduate students and people who are interested in directing projects for the participating students. All are welcome and encouraged to attend.

Much of the appeal of Thurston's work on the geometrization of 3-manifolds lies in its hands-on simplicity: synthetic and combinatorial constructions involving geodesic loops, simplicial surfaces, and coarse "quasi-geodesics" gain their power from rigidity theorems due to Mostow and Sullivan that guarantee that rough geometric estimates ensure spot-on control. But considerable analytic work of Ahflors and Bers lies at the foundation, and only recently has work of Graham and Witten been found to give clues to an analytic framework for understanding Thurston's intuition and conjectures. In this talk I will describe history context and recent developments that use Graham and Witten's notion of "renormalized volume", as elaborated by Krasnov and Schlenker, to provide a satisfying analytic explanation for the connection between volumes of hyperbolic 3-manifolds that fiber over the circle and Weil-Petersson lengths of closed geodesics on moduli space. I'll discuss an array of applications to Weil-Petersson geometry as well as some new results. This talk describes joint work with Ken Bromberg and Martin Bridgeman.

In the study of dynamical systems, mathematicians attempt to understand the long term behavior of systems which evolve in time by a mathematical set of rules. A polygon in the plane determines such a system via idealized billiard trajectories. The study of billiards in polygons gives rise to beautiful connections between the fields of dynamical systems, geometry, algebra, and number theory. In this talk, I will describe some of the motivating questions in the study of billiards, some of the celebrated results in this area, as well as some of the connections to other areas of mathematics.

We give a generalization to stacks of the classical (1920's) theorem of Petri -- we give a presentation for the canonical ring of a stacky curve. This is motivated by the following application: we give an explicit presentation for the ring of modular forms for a Fuchsian group with cofinite area, which depends on the signature of the group. This is joint work with John Voight.

The fundamental group is a more or less complete invariant of a 3-dimensional manifold. We will discuss how the purely algebraic question of whether or not this group has a left-invariant total order appears to be related to two other, seemingly quite different, properties of the manifold, one geometric and the other essentially analytic.

We study the geometry of horospheres in Teichmuller space of Riemann surfaces of genus g with n punctures, where 3g−3+n≥2. We show that every C^1-diffeomorphism of Teichmuller space to itself that preserves horospheres is an element of the extended mapping class group. Using the relation between horospheres and metric balls, we obtain a new proof of Royden's Theorem that the isometry group of the Teichmuller metric is the extended mapping class group. The work is joint with Dong Tan.

The classical hierarchy of Toda flows can be thought of as an action of the abelian group of polynomials on Jacobi matrices. We present a generalizations of this flow to the larger groups C^2 and entire functions, and we prove that the latter generalization remains an isospectral flow. This is joint work with Christian Remling.

The optimal transport problem defines a notion of distance in the space of probability measures over a manifold, the *Wasserstein space*. In his 1994 Ph.D. thesis, McCann discovered that this space is a length space: the distance between probability measures is given by the length of minimizing geodesics called *displacement interpolants*. A number of important functionals in physics and geometry turned out to be geodesically convex; their gradient flows give rise to nonlinear degenerate diffusion equations. In contrast with classical function spaces, the Wasserstein space is not a linear space, but rather an infinite-dimensional analogue of a Riemannian manifold. In this talk, I will describe how to differentiate functionals along displacement interpolants, using an Eulerian formulation for the underlying optimal transportation problem. Time permitting, I will sketch how to justify the calculations under minimal regularity assumptions, discuss geometric implications, and mention open questions. (Joint work with Benjamin Schachter)